Robust transitivity in hamiltonian dynamics
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 2, pp. 191-239.

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce C r open sets (r=1,2,,) of symplectic diffeomorphisms and Hamiltonian systems, exhibiting large robustly transitive sets. We show that the C closure of such open sets contains a variety of systems, including so-called a priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of the proof is a combination of studying minimal dynamics of symplectic iterated function systems and a new tool in Hamiltonian dynamics which we call “symplectic blender”.

Un objectif de ce travail est d’étudier la dynamique sur le complémentaire des tores KAM en mettant l’accent sur la transitivité robuste non locale. Nous introduisons les ensembles ouverts de difféomorphismes symplectiques et de systèmes hamiltoniens, présentant de grands ensembles robustement transitifs. L’adhérence de ces ensembles ouverts (en topologie C r , r=1,2,,) contient un grand nombre de systèmes, y compris les systèmes intégrables a priori instables. En outre, l'existence de mesures ergodiques avec un grand support est obtenue pour l'ensemble de ces systèmes. L'ingrédient principal des preuves est la combinaison de l'étude de systèmes itérés de fonctions de dynamique minimale et d'un nouvel outil de la dynamique hamiltonienne que nous appelons « mélangeurs symplectiques ».

DOI: 10.24033/asens.2164
Classification: 37D30, 37J40, 53Dxx, 70Fxx, 70Hxx
Keywords: symplectic blender, robust transitivity, hamiltonian dynamics, instability problem
Mot clés : mélangeurs symplectiques, transitivité robuste, dynamique hamiltonienne, problème d'instabilité
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Nassiri, Meysam; Pujals, Enrique R. Robust transitivity in hamiltonian dynamics. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 2, pp. 191-239. doi : 10.24033/asens.2164. http://www.numdam.org/articles/10.24033/asens.2164/

[1] F. Abdenur, C. Bonatti & S. Crovisier, Nonuniform hyperbolicity for C 1 -generic diffeomorphisms, Israel J. Math. 183 (2011), 1-60. | MR | Zbl

[2] L. Arnold, Random dynamical systems, Springer Monographs in Math., Springer, 1998. | MR | Zbl

[3] V. I. ArnolʼD, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), 91-192. | MR | Zbl

[4] V. I. ArnolʼD, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR 156 (1964), 9-12. | MR | Zbl

[5] V. I. Arnold, V. V. Kozlov & A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, third éd., Encyclopaedia of Math. Sciences 3, Springer, 2006. | MR | Zbl

[6] D. Bernstein & A. Katok, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians, Invent. Math. 88 (1987), 225-241. | MR | Zbl

[7] C. Bonatti & L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. 143 (1996), 357-396. | MR | Zbl

[8] C. Bonatti & L. J. Díaz, Robust heterodimensional cycles and C 1 -generic dynamics, J. Inst. Math. Jussieu 7 (2008), 469-525. | MR | Zbl

[9] C. Bonatti, L. J. Díaz & E. R. Pujals, A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. 158 (2003), 355-418. | MR | Zbl

[10] C. Bonatti, L. J. Díaz & M. Viana, Dynamics beyond uniform hyperbolicity, Encyclopedia of Mathematical Sciences, Springer, 2004. | Zbl

[11] K. Burns & A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. 171 (2010), 451-489. | MR | Zbl

[12] C.-Q. Cheng & J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom. 67 (2004), 457-517. | MR | Zbl

[13] A. Delshams, M. Gidea, R. De La Llave & T. M. Seara, Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation, in Hamiltonian dynamical systems and applications, NATO Sci. Peace Secur. Ser. B Phys. Biophys., Springer, 2008, 285-336. | MR | Zbl

[14] A. Delshams, R. De La Llave & T. M. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model, Mem. Amer. Math. Soc. 179 (2006). | Zbl

[15] A. Delshams, R. De La Llave & T. M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math. 202 (2006), 64-188. | MR | Zbl

[16] L. J. Díaz, E. R. Pujals & R. Ures, Partial hyperbolicity and robust transitivity, Acta Math. 183 (1999), 1-43. | MR | Zbl

[17] R. Douady, Stabilité ou instabilité des points fixes elliptiques, Ann. Sci. École Norm. Sup. 21 (1988), 1-46. | Numdam | MR | Zbl

[18] F. H. Ghane, A. J. Homburg & A. Sarizadeh, C 1 robustly minimal iterated function systems, Stoch. Dyn. 10 (2010), 155-160. | MR | Zbl

[19] M. W. Hirsch, C. Pugh & M. Shub, Invariant manifolds, Lecture Notes in Math. 583, Springer, 1977. | MR | Zbl

[20] V. Horita & A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 641-661. | Numdam | MR | Zbl

[21] V. Kaloshin & M. Levi, An example of Arnold diffusion for near-integrable Hamiltonians, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 409-427. | MR | Zbl

[22] V. Kaloshin, J. N. Mather & E. Valdinoci, Instability of resonant totally elliptic points of symplectic maps in dimension 4, Astérisque 297 (2004), 79-116. | Numdam | MR | Zbl

[23] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. I.H.É.S. 51 (1980), 137-173. | Numdam | MR | Zbl

[24] A. Koropecki & M. Nassiri, Transitivity of generic semigroups of area-preserving surface diffeomorphisms, Math. Z. 266 (2010), 707-718; 268 (2011), 601-604. | MR | Zbl

[25] R. Mañé, Contributions to the stability conjecture, Topology 17 (1978), 383-396. | MR | Zbl

[26] R. Mañé, Ergodic theory and differentiable dynamics, Ergebnisse Math. Grenzgb. 8, Springer, 1987. | MR | Zbl

[27] J.-P. Marco & D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems, Publ. Math. I.H.É.S. 96 (2002), 199-275. | Numdam | MR | Zbl

[28] J. N. Mather, Arnolʼd diffusion. I. Announcement of results, Sovrem. Mat. Fundam. Napravl. 2 (2003), 116-130; English transl.in J. Math. Sci. 124 (2004), 5275-5289. | MR | Zbl

[29] R. Moeckel, Generic drift on Cantor sets of annuli, in Celestial mechanics (Evanston, IL, 1999), Contemp. Math. 292, Amer. Math. Soc., 2002, 163-171. | MR | Zbl

[30] M. Nassiri, Robustly transitive sets in nearly integrable Hamiltonian systems, Thèse, IMPA, 2006.

[31] S. E. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Amer. J. Math. 99 (1977), 1061-1087. | MR | Zbl

[32] C. Pugh & M. Shub, Stable ergodicity, Bull. Amer. Math. Soc. (N.S.) 41 (2004), 1-41. | MR

[33] E. R. Pujals & M. Sambarino, Homoclinic bifurcations, dominated splitting, and robust transitivity, in Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, 327-378. | MR | Zbl

[34] R. C. Robinson, Generic properties of conservative systems I, II, Amer. J. Math. 92 (1970), 562-603, 897-906. | MR | Zbl

[35] R. Saghin & Z. Xia, Partial hyperbolicity or dense elliptic periodic points for C 1 -generic symplectic diffeomorphisms, Trans. Amer. Math. Soc. 358 (2006), 5119-5138. | MR | Zbl

[36] M. Shub, Topologically transitive diffeomorphisms of 𝕋 4 , in Symposium on Differential Equations and Dynamical Systems, Springer Lecture Notes 206, 1971, 39-40.

[37] Z. Xia, Arnold diffusion: a variational construction, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Extra Vol. II, 1998, 867-877. | MR | Zbl

[38] E. Zehnder, Homoclinic points near elliptic fixed points, Comm. Pure Appl. Math. 26 (1973), 131-182. | MR | Zbl

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